1. a) An (8,4) code C is defined by the mapping abcd abcdwxyz where w= a + b + c (mod 2), x= a + b + d (mod 2), y = a +c+d (mod 2), z = b + c + d (mod 2) i) Write down generator and check matrices, G and H, for this code. (6 marks) ii) What is the minimum distance for this code? (3 marks) iii) Is it possible for C to correct 1 error in a received word and at the same time detect a second error? Explain your answer quoting any relevant theorems. (2 marks) iv) Is it possible to add an additional check equation to C that would allow us to correct 2 errors in a received word? (3 marks) b) A different (7,3) code C' is defined by the check matrix H . 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 i) Identify the values for (a), (b) and (c) in the following coset leader/syndrome table for C Coset Leader 0000000 1000000 0100000 0010000 0001000 0000100 0000010 0000001 1000001 (a) 0001010 0001001 0000110 0000101 0000011 Syndrome 0000 1101 1011 0111 1000 0100 0010 0001 1110 1100 1010 (b) 0110 0101 0011 1111 (3 marks) ii) Use the table to find a codeword a smallest distance from the received word w=1001110. (3 marks) 1. a) An (8,4) code C is defined by the mapping abcd abcdwxyz where w= a + b + c (mod 2), x= a + b + d (mod 2), y = a +c+d (mod 2), z = b + c + d (mod 2) i) Write down generator and check matrices, G and H, for this code. (6 marks) ii) What is the minimum distance for this code? (3 marks) iii) Is it possible for C to correct 1 error in a received word and at the same time detect a second error? Explain your answer quoting any relevant theorems. (2 marks) iv) Is it possible to add an additional check equation to C that would allow us to correct 2 errors in a received word? (3 marks) b) A different (7,3) code C' is defined by the check matrix H . 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 1 i) Identify the values for (a), (b) and (c) in the following coset leader/syndrome table for C Coset Leader 0000000 1000000 0100000 0010000 0001000 0000100 0000010 0000001 1000001 (a) 0001010 0001001 0000110 0000101 0000011 Syndrome 0000 1101 1011 0111 1000 0100 0010 0001 1110 1100 1010 (b) 0110 0101 0011 1111 (3 marks) ii) Use the table to find a codeword a smallest distance from the received word w=1001110